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Monotone Optimisation with Learned Projections

Ahmed Rashwan, Keith Briggs, Chris Budd, Lisa Kreusser

TL;DR

This work addresses global monotone optimisation when objective and constraint functions are available only via data. It proposes learning the radial inverse, the projection primitive used by the Polyblock Outer Approximation (POA) algorithm, and introduces Homogeneous–Monotone Radial Inverse (HM-RI) networks to estimate these projections while preserving monotonicity and positive homogeneity. By aligning learning targets with POA’s internal structure, the approach achieves significant runtime speed-ups and near-global solutions across indefinite quadratic programming, multiplicative programming, and transmit power optimisation tasks, outperforming non-monotone baselines. Relaxed monotonicity and simplified HM-RI variants further improve training efficiency and practical performance. Overall, radial-inverse learning offers a principled, scalable path for integrating data-driven components into global monotone optimisation pipelines, with strong empirical validation and clear engineering advantages.

Abstract

Monotone optimisation problems admit specialised global solvers such as the Polyblock Outer Approximation (POA) algorithm, but these methods typically require explicit objective and constraint functions. In many applications, these functions are only available through data, making POA difficult to apply directly. We introduce an algorithm-aware learning approach that integrates learned models into POA by directly predicting its projection primitive via the radial inverse, avoiding the costly bisection procedure used in standard POA. We propose Homogeneous-Monotone Radial Inverse (HM-RI) networks, structured neural architectures that enforce key monotonicity and homogeneity properties, enabling fast projection estimation. We provide a theoretical characterisation of radial inverse functions and show that, under mild structural conditions, a HM-RI predictor corresponds to the radial inverse of a valid set of monotone constraints. To reduce training overhead, we further develop relaxed monotonicity conditions that remain compatible with POA. Across multiple monotone optimisation benchmarks (indefinite quadratic programming, multiplicative programming, and transmit power optimisation), our approach yields substantial speed-ups in comparison to direct function estimation while maintaining strong solution quality, outperforming baselines that do not exploit monotonic structure.

Monotone Optimisation with Learned Projections

TL;DR

This work addresses global monotone optimisation when objective and constraint functions are available only via data. It proposes learning the radial inverse, the projection primitive used by the Polyblock Outer Approximation (POA) algorithm, and introduces Homogeneous–Monotone Radial Inverse (HM-RI) networks to estimate these projections while preserving monotonicity and positive homogeneity. By aligning learning targets with POA’s internal structure, the approach achieves significant runtime speed-ups and near-global solutions across indefinite quadratic programming, multiplicative programming, and transmit power optimisation tasks, outperforming non-monotone baselines. Relaxed monotonicity and simplified HM-RI variants further improve training efficiency and practical performance. Overall, radial-inverse learning offers a principled, scalable path for integrating data-driven components into global monotone optimisation pipelines, with strong empirical validation and clear engineering advantages.

Abstract

Monotone optimisation problems admit specialised global solvers such as the Polyblock Outer Approximation (POA) algorithm, but these methods typically require explicit objective and constraint functions. In many applications, these functions are only available through data, making POA difficult to apply directly. We introduce an algorithm-aware learning approach that integrates learned models into POA by directly predicting its projection primitive via the radial inverse, avoiding the costly bisection procedure used in standard POA. We propose Homogeneous-Monotone Radial Inverse (HM-RI) networks, structured neural architectures that enforce key monotonicity and homogeneity properties, enabling fast projection estimation. We provide a theoretical characterisation of radial inverse functions and show that, under mild structural conditions, a HM-RI predictor corresponds to the radial inverse of a valid set of monotone constraints. To reduce training overhead, we further develop relaxed monotonicity conditions that remain compatible with POA. Across multiple monotone optimisation benchmarks (indefinite quadratic programming, multiplicative programming, and transmit power optimisation), our approach yields substantial speed-ups in comparison to direct function estimation while maintaining strong solution quality, outperforming baselines that do not exploit monotonic structure.
Paper Structure (52 sections, 4 theorems, 51 equations, 1 figure, 4 tables, 2 algorithms)

This paper contains 52 sections, 4 theorems, 51 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Monotone optimisation
  5. Polyblocks and outer approximation
  6. Rectangles and polyblocks.
  7. Monotone projection.
  8. Polyblock Outer Approximation (POA).
  9. Learning monotone constraints
  10. Learned objective and constraints.
  11. Certified monotone networks.
  12. Estimating projections using learned radial inverse functions
  13. Radial inverse function
  14. From radial inverse functions to monotone projections
  15. Structured models for radial inverse estimation
  16. ...and 37 more sections

Key Result

Proposition 1

Let $g:\mathbb{R}_+^n\to\mathbb{R}$ be an increasing function, and let $x, x'\in \mathbb R^n_+$ and $y, y'\in \mathbb R$. Then $\rho_g$ satisfies:

Figures (1)

  • Figure 1: Illustration of the construction of $V^0=\{b\}, V^1=\{V^1_1,V^1_2\}$ and $V^2 =\{V^2_1,V^2_2, V^2_3\}$ of POA in 2D with $m_h=0$, i.e., $\mathcal{H} =\mathbb R^2_+$. Points are color coded by iteration. Vertices are indexed by subscripts.

Theorems & Definitions (7)

  • Proposition 1: Radial inverse properties
  • Proposition 2: Projection via radial inverses
  • Proposition 3
  • proof : Proof of Proposition \ref{['prop:h_properties']}
  • proof : Proof of Proposition \ref{['prop:proj_via_ri']}
  • proof : Proof of Proposition \ref{['prop:inverse']}
  • Theorem 1